Deconfinement of Mott localized electrons into topological and spin–orbit-coupled Dirac fermions

The interplay of electronic correlations, spin–orbit coupling and topology holds promise for the realization of exotic states of quantum matter. Models of strongly interacting electrons on honeycomb lattices have revealed rich phase diagrams featuring unconventional quantum states including chiral superconductivity and correlated quantum spin Hall insulators intertwining with complex magnetic order. Material realizations of these electronic states are, however, scarce or inexistent. In this work, we propose and show that stacking 1T-TaSe2 into bilayers can deconfine electrons from a deep Mott insulating state in the monolayer to a system of correlated Dirac fermions subject to sizable spin–orbit coupling in the bilayer. 1T-TaSe2 develops a Star-of-David charge density wave pattern in each layer. When the Star-of-David centers belonging to two adyacent layers are stacked in a honeycomb pattern, the system realizes a generalized Kane–Mele–Hubbard model in a regime where Dirac semimetallic states are subject to significant Mott–Hubbard interactions and spin–orbit coupling. At charge neutrality, the system is close to a quantum phase transition between a quantum spin Hall and an antiferromagnetic insulator. We identify a perpendicular electric field and the twisting angle as two knobs to control topology and spin–orbit coupling in the system. Their combination can drive it across hitherto unexplored grounds of correlated electron physics, including a quantum tricritical point and an exotic first-order topological phase transition.


INTRODUCTION
Prospects of quantum information technologies have motivated an intense search for systems, which intertwine topology and electronic correlations [1][2][3][4] . Strongly interacting and spin-orbitcoupled electrons on the honeycomb lattice, as theoretically described by the Kane-Mele-Hubbard model 5 , feature quantum spin Hall (QSH), Mott-Hubbard, collective magnetic and chiral superconducting states. While several weakly interacting QSH systems are now well-studied experimentally 2,6 , material realizations of honeycomb Kane-Mele-Hubbard fermions with strong correlations and spin-orbit coupling are rare 7,8 . Here, we introduce a new "van der Waals engineering" platform to serve this purpose.
We show that stacking of 1T-TaSe 2 into bilayers can deconfine electrons from a deep Mott insulating state realized in the monolayer to a system of correlated Dirac fermions subject to sizable spin-orbit coupling. Central to this transition is the possibility of van der Waals materials to stack in different configurations. For a specific honeycomb arrangement (Fig. 1) the kinetic energy associated with the electronic hopping t turns out to be of the same order of magnitude as the effective local Coulomb repulsion U. The system features, therefore, electronic correlations, which turn out to put the system right on the verge between QSH and correlated antiferromagnetic insulating states at charge neutrality and support chiral superconductivity under doping. We finally demonstrate that tuning the system via electric fields and twisting of the layers relative to each other sensitively affects the low-energy electronic structure in terms of emerging Dirac mass and spin-orbit coupling terms and leads to completely unexplored regimes of correlated electrons.

RESULTS AND DISCUSSION
From a correlated insulator to emergent Dirac fermions Layered group-V transition metal dichalcogenides (TMDCs) such as 1T-TaSe 2 or 1T-TaS 2 feature a low-temperature commensurate charge density wave (CCDW) where Ta-atoms are displaced into Star-of-David (SoD) patterns ( Fig. 1a) 9,10 . In this phase, the SoDs form a triangular ffiffiffiffiffi 13 p ffiffiffiffiffi 13 p superlattice in every layer and host correlated electrons: 1T-TaS 2 shows a metal-to-insulator transition when entering the CCDW phase 10 . In 1T-TaSe 2 , the bulk remains conductive till lowest temperatures, while the surface exhibits a Mott transition around 250 K 11,12 . Recently, 1T-TaSe 2 has been fabricated down to monolayer thickness [13][14][15] and a pronounced thickness dependence of the electronic structure has been reported 15 . As bonds between the layers are mainly of van der Waals type, different stacking configurations are observed in experiments 16,17 and have a strong impact on the electronic structure [18][19][20] .
We compare the CCDW state in the monolayer to a bilayer with honeycomb stacking, where the SoD centers form a buckled honeycomb lattice (Fig. 1b). The bottom layer SoD centers form sublattice A and the top layer sublattice B. This stacking is one of many possible configurations, which can generally differ, both, by the local atomic stacking and by the stacking of the SoD centers. The local stacking considered here, has the Ta atoms of the bottom layer approximately beneath the lower Se atoms of the top layer (Fig. 1c). While this kind of local stacking is not the most commonly observed one of the T-phase TMDCs 15,17 , a corresponding stacking sequence has been found in transmission electron microscopy studies of 1T-TaS 2 17 and turns out to be metastable in density functional theory (DFT) simulations of 1T-TaSe 2 21 . Regarding the stacking of the SoD centers further van der Waals DFT total energy calculations (see Supplementary Information) show that the particular honeycomb arrangement studied here, is metastable and energetically on the order of 10 meV per formula unit above the lowest energy configuration. This is similar to the configuration observed experimentally in ref. 15 and it is thus plausible that also the honeycomb configuration considered here, is within reach of experiments. Experimental approaches including tear-and-stack 22 and Scanning Tunneling Microscopy voltage pulse-based manipulation schemes 23 can present possible avenues to control and switch metastable stacking configurations and to reach the honeycomb configurations discussed in our paper. At small twist angles all different kinds of configurations can be realized locally in the moiré, including likely those honeycomb cases discussed here.
To study the electronic structure of such engineered stackings, we combine ab-initio calculations in the frameworks of DFT and the random phase approximation (RPA) with effective low-energy models, which we investigate with dynamical mean-field theory (DMFT) and two-particle self-consistent (TPSC) many-body approaches, see "Methods" section.
In the CCDW phase, the band structure of monolayer 1T-TaSe 2 obtained with non-spin-polarized DFT is characterized by a single (Ta) flat band at the Fermi level 10,18-20 , which has a bandwidth of less than 20 meV (Fig. 2a, left). Hence, the CCDW formation largely quenches in-plane hopping of the electrons. In the honeycombstacked bilayer (with no twist), two dispersive bands with a bandwidth of the order of 200 meV emerge from the low-energy flat band of the monolayer (Fig. 2a, right). Comparison of the mono-and bilayer bandwidths shows that interlayer hopping effects must dominate over intralayer hoppings by approximately an order of magnitude. In this sense, CCDW TaSe 2 bilayers are the exact opposite of graphene bilayer systems, since in the latter outof-plane coupling is an order of magnitude weaker than in-plane hopping 24,25 . For the 1T-TaSe 2 honeycomb bilayer, the upper and lower low-energy bands touch as Dirac points at the Brillouin zone corners K and K'. In the undoped system, these Dirac points are exactly at the Fermi level.
We next construct a Wannier Hamiltonian to describe the Dirac bands with one Wannier function for each SoD center, i.e., two Wannier orbitals per bilayer CCDW superlattice unit cell (Supplemental Fig. 2). The resulting nearest-neighbor hopping between a sublattice A site in the bottom and a neighboring sublattice B site in the top layer amounts to t = −34 meV and is the leading term of the Wannier Hamiltonian. There are further terms in the Wannier Hamiltonian, which are, however, at least an order of magnitude smaller than t.
The effective Hubbard interaction U for the SoD Wannier orbitals of CCDW TaSe 2 calculated in RPA is U ≈ 130 meV, which is also in line with the experimental estimates in ref. 15 and calculations for TaS 2 19 . The ratio of hopping to Coulomb interaction is decisive in determining the strength and kind of electronic correlation phenomena taking place. Our calculations yield U/|t| ≈ 3.8.
To study the resulting electronic correlations, we performed simulations of the Hubbard model for the non-twisted CCDW 1T-TaSe 2 bilayer in the framework of DMFT and the TPSC approach 26,27 . The quasi-particle weight Z shown in Fig. 2c is a measure of the electronic correlation strength. In the temperature range T = 60-230 K, both DMFT and TPSC consistently yield essentially constant Z ≈ 0.75. Our system is thus at intermediate local correlation strength and far away from the paramagnetic Mott-Hubbard transition taking place above U Mott /t ≈ 8.2. The DMFT and TPSC studies of the full Hubbard model for the nontwisted CCDW 1T-TaSe 2 bilayer are, thus, in line with studies of the idealized Hubbard model on the honeycomb lattice 28 . The TPSC calculations give insight to spin-fluctuations taking place in the system (Supplemental Fig. 4). The inverse intra-and inter-sublattice terms of the static magnetic susceptibility at wave number q = 0, as well as the antiferromagnetic correlation length ξ AFM are shown in Fig. 2d. We observe that antiferromagnetic fluctuations with alternating spin orientation between the two sublattices (A, B) are dominant and strongly enhanced at temperatures T ≲ 100 K. These fluctuations indicate that the nontwisted CCDW 1T-TaSe 2 bilayer is close to a quantum phase transition from a Dirac semimetal to an antiferromagnetic insulator, which occurs for ideal Hubbard honeycomb systems exactly in the range of U c /t ≈ 3.6-3.8 6,29,30 .
Spin-orbit coupling The aforementioned magnetic correlation phenomena are sensitive to details of the low-energy electronic structure and spin-orbit coupling (SOC), which we discuss in the following based on a symmetry analysis. The space group of non-twisted CCDW 1T-TaSe 2 bilayer in the honeycomb structure is P3 (#147), comprising inversion symmetry and three-fold rotations C 3 around an axis perpendicular to the bilayer. Imposing also time-reversal symmetry, every band must be two-fold degenerate. Therefore, SOC-induced qualitative changes of the band structures can occur near the Dirac points at K and K′. A corresponding k ⋅ p expansion (see Supplementary Information) reads where the pseudospin S describes the sublattice degree of freedom, σ acts on the electron spin, and τ = ±1 labels the valley (K, K′) degree of freedom. This Hamiltonian comprises three contributions; the first contribution is a two-dimensional massless Dirac term, with the sublattice-pseudospin playing the role of the spin inherent to the Dirac equation. This term is analogous to the massless Dirac term in graphene 6,24,25 . SOC is responsible for the second and third contributions to H 0 : a valley-spin-sublattice coupling λ SOC = 0.74 meV, which is often called Kane-Mele spinorbit term 31,32 , and a sublattice-staggered Rashba term α R2 , which belongs to the R2 class according to the classification form ref. 33 . A finite Kane-Mele term λ SOC opens a gap and turns the system described by H 0 into a QSH insulator 31,32 . Importantly, the Kane-Mele term here is enhanced in comparison to its counterpart in graphene by two orders of magnitude 24,25,34 , and corresponds to a temperature T ≈ 10 K, which is well accessible in experiments. Given that U/|t| ≈ 3.8, the non-twisted CCDW 1T-TaSe 2 bilayer implements a material realization of the Kane-Mele-Hubbard model in a regime very close to the topological quantum phase transition from a QSH insulator to an antiferromagnetic insulator (Fig. 2e). Dirac fermions and their topology are affected by different kinds of mass fields. An energy difference between electrons localized in sublattice A and B leads to a so-called Semenoff mass term M, which would enter the Hamiltonian H 0 of equation (1) in the form MS z . This term breaks sublattice invariance and thereby inversion symmetry and leads to a transition from a QSH to a band insulator at |M| = |λ SOC | 31, 35 . In the non-twisted CCDW 1T-TaSe 2 bilayer, the intrinsic Semenoff mass M 0 = 0 is required to vanish by Influence of extrinsic Semenoff mass terms ΔM on the low-energy band structure. The sublattice character is color coded. The system changes from QSH to trivial band insulator at ΔM = 0.82 meV, which corresponds to a vertical electric field of E z ≈ 0.5 mV Å −1 . c Quasi-particle weight Z for non-twisted CCDW 1T-TaSe 2 bilayer calculated with DMFT and TPSC. Both approaches place the system consistently in the moderately correlated regime Z ≈ 0.75 at all calculated temperatures. d Temperaturedependent antiferromagnetic correlation length ξ AFM and inverse static spin susceptibilities of non-twisted CCDW 1T-TaSe 2 bilayer at wave vector q = 0 as calculated with TPSC. The intra-sublattice (1/χ AA = 1/χ BB ) and inter-sublattice (1/χ AB ) elements of the inverse susceptibility at wave number q = 0 are shown. e Schematic phase diagram of honeycomb-stacked non-twisted CCDW 1T-TaSe 2 bilayer as a function of extrinsic Semenoff mass ΔM and interaction strength U. The region accessible for non-twisted CCDW 1T-TaSe 2 bilayer through tuning with external electric fields is highlighted. The transition from QSH to band insulator is a continuous transition at small U (dashed line) and a firstorder transition at larger U (solid line). The red area in the quantum spin Hall region indicates the increasing many-body character of this phase.
symmetry. Our Wannier construction yields M 0 = 0.14 meV, which is small as compared to all other relevant terms in the system. Possible origins of this small symmetry breaking can be the Wannier constructions and also faint asymmetries accumulated during self-consistency iterations in the DFT calculations. Vertical electric fields E z , as realizable in field effect transistor geometries (Fig. 1c), break inversion symmetry, translate into staggered sublattice potentials, and therefore corresponding extrinsic Semenoff mass contributions ΔM (Fig. 2b). DFT calculations (Supplementary Fig. 3) yield the approximate relation ΔM ≈ eE z d/ϵ ⊥ , where d = 6.4 Å is the interlayer distance, e is the elementary charge, and ϵ ⊥ = 3.72 plays the role of an effective dielectric constant. The QSH to band insulator transition is reached for E z ≈ 0.5 mV Å −1 = 50 kV cm −1 (Fig. 2b), which is well within reach of experiments 36 .
Taken together, our calculations show that the honeycombstacked non-twisted CCDW 1T-TaSe 2 bilayer is located in a region of the phase diagram (Fig. 2e) with three different phases (QSH insulator, band insulator and antiferromagnetic insulator) coming together. Electron correlation is known to change the order of the QSH-band insulator transition from second to first-order 37 . Contrary to the standard non-interacting QSH to band insulator transition, where the gap closes and reopens continuously with vanishing gap at the transition point, the QSH gap remains finite and the system changes discontinuously to a band insulating state at the transition point. Application of vertical electric fields in the system at hand represents hence a possibility to realize this exotic interaction-induced first-order transition.
Twisted CCDW 1T-TaSe 2 bilayers Layered van der Waals systems allow to realize different stacking configurations via twisting, i.e., relative rotations between the layers. General twist angles θ lead to incommensurate moiré patterns superimposed to the CCDW lattice. Arguably the simplest case of twisting is a rotation angle of θ = 180 ∘ , which leads to a system with identical Bravais lattice but different symmetry of the supercell basis (Fig. 1c): in case of honeycomb stacking of the CCDW, the space group of the 180 ∘ twisted structure of CCDW TaSe 2 bilayer is reduced to P3, meaning that inversion symmetry is lost with respect to the non-twisted case. The resulting band structure (Fig. 3a) is qualitatively similar to the non-twisted honeycomb case regarding the overall shape and width of the low-energy bands. Thus, also the 180 ∘ twisted case will be far away from the paramagnetic Mott transition. However, the low-energy band structure is markedly different in the 180 ∘ twisted case. First, the conduction band is almost flat between K and M. Second, inversion symmetry breaking lifts band degeneracies: our DFT calculations reveal a staggered potential and an associated intrinsic Semenoff mass term of M 0 = 8.55 meV, which opens a gap at the K and K' points already without SOC and without external electric fields. Furthermore, additional SOC terms are now allowed by symmetry (see Supplementary Information) and completely lift the remaining band degeneracies except for the time-reversal symmetric points Γ and M.
Based on a symmetry analysis, we obtain the following lowenergy model in the vicinity of K and K′: The additional terms with respect to the non-twisted honeycomb structure equation (1)   (with coupling constant λ D ) can also be seen as an effective k-dependent magnetic field parallel to the z-axis. Our DFT calculations yield M 0 = 8.55 meV, B = −1.85 meV, |λ SOC | ≲ 0.05 meV and λ R = 3.21 meV. These terms affect the dispersion and imprint an intricate sublattice and spin structure to the lowenergy bands, which can be manipulated by vertical electric fields as shown in Fig. 3b.
Except for E z = −4.2 meV Å −1 (ΔM = −6.68 meV and M = M 0 + ΔM = 1.87 meV), the system is always gapped. We calculated the Z 2 topological invariant for the non-interacting 180 ∘ twisted bilayer in comparison to the non-twisted bilayer case,, as well as for two cases in between where the ratio B/λ SOC is varied (Fig. 3c, and Supplemental Information). In the non-twisted bilayer, the system is in a QSH state unless an extrinsic sufficiently large Semenoff mass term or an additional Rashba SOC term λ R are added. In the 180 ∘ twisted case, the situation is very different regardless whether or not the intrinsic Semenoff mass term is compensated by an external electric field and regardless of λ SOC . Indeed, many changes in the SOC terms suppress the QSH state in the 180 ∘ twisted bilayer: The comparably large Rashba λ R and the spin-valley coupling terms B and a strong reduction in λ SOC . Each of these alone is sufficient to suppress the QSH state. At vertical electric field E z = −4.2 mV Å −1 the gaps at K and K' close, and a band touching point emerges.
While it is clear that there will be tendencies towards interaction-induced (quasi)ordered phases as well, here, the kind of ordering is likely different from the non-twisted case but largely unexplored. The band touching at E z = −4.2 mV Å −1 implements a situation similar to saddle points in a two-dimensional dispersion, where already arbitrarily weak interactions would trigger different kinds of magnetic or excitonic instabilities 38 . How these instabilities translate into the intermediately correlated and strongly spin-orbit-coupled case of 180 ∘ twisted TaSe 2 is a completely open question.

Conclusions and outlook
The field of twistronics with materials like bilayer graphene is based on the idea that weak interlayer coupling can flatten highly dispersive bands and thereby boost electronic correlations [39][40][41] . The system introduced here takes the opposite route of deconfining formerly Mott localized electrons. This approach should be generally applicable to interfaces of Mott localized electrons under two conditions: the interlayer coupling should substantially exceed the in-plane one and at the same time define a connected graph linking all sites of the system. Possible example systems range from stacking faults in the bulk of CCDW layered Mott materials 17 to molecular systems 42 .
Especially the twisting degree of freedom opens new directions to experiments. Since interlayer hopping is the dominant kinetic term in deconfined Mott systems like bilayers of CCDW 1T-TaSe 2 , we expect incommensurability effects to be much more pronounced than in twisted graphene systems 39 . θ = 30 ∘ twisted CCDW 1T-TaSe 2 bilayer should realize a quasicrystal with twelvefold rotation symmetry and provide an experimental route to correlated electrons and emerging collective states in a quasicrystalline environment.
The prototypical case of CCDW 1T-TaSe 2 bilayer demonstrates how deconfinement of Mott electrons leads to exotic states of quantum matter: the non-twisted bilayer approaches a quantum tricritical region of competing QSH, trivial band insulating, and antiferromagnetic insulating states. At 180 ∘ twist angle different kinds of electrically controllable band degeneracies with associated many-body instabilities, hypothetically of excitonic type, emerge. Clearly, the phase space for manipulating deconfined Mott electrons is high dimensional. We here identified the combination of twist angle and perpendicular electric field as decisive for TaSe 2 bilayers. Further means to control emerging electronic states include dielectric engineering 43 and charge doping. Our calculations showed that the non-twisted CCDW 1T-TaSe 2 bilayer in honeycomb stacking approximates the (Kane-Mele) Hubbard model on the honeycomb lattice with U/|t| ≈ 3.8 very well. In this regime, doping the system away from the Dirac point towards the van Hove singularity is expected to lead to chiral superconductivity [44][45][46][47] , most likely of d + id-type.

DFT calculations
We perform DFT 48,49 calculations by using the Vienna ab-initio simulation package (VASP) 50,51 with the generalized gradient approximation of Perdew, Burke, and Ernzerhof (GGA-PBE) for the exchange-correlation functional 52,53 . We obtain the total energies, relaxed structures and electronic structure of mono-and bilayer 1T-TaSe 2 systems. We calculate the total energies for various possible stackings in undistorted and CCDW bilayer 1T-TaSe 2 using Γ-centered k-meshes of 15 × 15 × 1 and 9 × 9 × 1, respectively, and taking into account van der Waals (vdW) corrections within DFT-D2 and cross-checking with DFT-D3 54,55 , see Supplemental Fig.  1. The ionic relaxation is done using the conjugate gradient algorithm until all force components are smaller than 0.02 eV Å −1 .
Since the DFT-D2 corrections yield correct interlayer distances but do not correctly capture the CCDW distortions, we adopted the following relaxation procedure to calculate the commensurate ffiffiffiffiffi 13 p ffiffiffiffiffi 13 p CCDW bilayer structures: 1. We perform relaxations for a ffiffiffiffiffi 13 p ffiffiffiffiffi 13 p supercell of the monolayer (without vdW corrections), using a superlattice constant of a = 12.63 Å according to ref. 15 . We fix the vertical positions of the Ta atoms, while allowing for Ta in-plane displacements. The Se atoms are allowed to freely relax in all three directions. 2. We include then a second layer and optimize the interlayer distance, d, while keeping all relative intralayer distances fixed. We find d = 6.4 Å for the ideal honeycomb stacking in CCDW bilayer 1T-TaSe 2 . 3. We relax the CCDW bilayer following the same procedure described for the monolayer, i.e., without vdW corrections and vertical positions of the Ta atoms according to the optimized interlayer distance d fixed, while allowing for in-plane displacements. The Se atoms are allowed to freely relax in all three directions. We cross check the results obtained by this procedure against calculations with vdW corrections according to the DFT-D3 method 55 . In contrast to DFT-D2, the CCDW distortions are well described in the DFT-D3. Thus, full relaxations of all atomic positions have been performed in the DFT-D3 framework for the mono-and bilayer. Both our step-by-step procedure using DFT-D2 method, and the full relaxation using DFT-D3 give equivalent results for the total energies (see Supplemental Fig. 1b), for the crystal and band structures.
For the non-collinear magnetic calculation, i.e., when SOC is included, we set the net magnetic moment to zero in all atoms of the unit cell, and use a Γ-centered k-mesh of 6 × 6 × 1.

Estimation of the screened Hubbard interaction U via RPA
We estimate the local Hubbard interaction U for the flat bands around the Fermi level in the CCDW 1T-TaSe 2 bilayer from the ab-initio calculation of the screened Coulomb interaction, using RPA for the undistorted bilayer. We follow a similar procedure as in ref. 56 , which we summarize below: • We initially calculate the WANNIER90 tight-binding model for the three low-energy Ta bands C, whose orbital character is mostly fd z 2 ; d x 2 Ày 2 ; d xy g (see Fig. 2(a) in ref. 56 ) in the undistorted monolayer 1T-TaSe 2 .

•
In the CCDW 1T-TaSe 2 bilayer, the d z 2 orbitals from Ta atoms in the SoD centers have the largest contribution for the bands around the Fermi level. Thus, for each q, we consider only the tensor element W (q) ≡ W αααα (q) with α ¼ d z 2 .
• Then, the local Hubbard interaction U in a single star of David is calculated by averaging over the d z 2 orbital weight from each Ta atom J.M. Pizarro et al.
(labeled by w d z 2 ðRÞ) in the star of David: where U(R) is the discrete Fourier transform of W(q).

DMFT and TPSC many-body calculations
For non-twisted CCDW 1T-TaSe 2 bilayer, we construct tight-binding-Hubbard Hamiltonians of the type H TBH ¼ X hi;ji;σ;σ 0 t σσ 0 ij c y iσ c jσ 0 þ U X i n i" n i# (4) from a Wannierization of the ab-initio DFT band structure (Supplementary Information), and estimate the on-site repulsion U to be about 130 meV by means of RPA calculations. We study these effective low-energy models from a many-body perspective to judge the type of correlations in the system. In DMFT the lattice Hamiltonian is mapped onto a selfconsistently determined single impurity problem, solved-in our casewithin numerical exact quantum Monte Carlo in the hybridization expansion flavor (CT-HYB) (for a review, see 57 ). The resulting sublatticeresolved self-energy, Σ, is local (k-independent) but it contains frequencydependent non-perturbative corrections beyond Hartree-Fock to all orders and can account for Mott-Hubbard metal-to-insulator transitions. All DMFT calculations are performed using w2dynamics 58 . The double counting is accounted for using the fully localized limit. For twodimensional systems it is important to estimate non-local effects at the level of the self-energy, not included in DMFT. To this goal, we apply the TPSC method 59 , which produces accurate results in the weak-tointermediate coupling regime, if compared to lattice quantum Monte Carlo calculations in the single band Hubbard model. For non-twisted CCDW 1T-TaSe 2 bilayer, which is modeled by a multi-band system we use the multi-site formulation of TPSC 26 while neglecting the Hartree term to avoid double counting of correlation effects already accounted for in DFT. Moreover, to be able to apply TPSC to this system we project out spin offdiagonal terms and take only the diagonal spin-up contributions from DFT while still assuming a paramagnetic state. The combination of TPSC accounting for the k-dependence of Σ and DMFT, in which we can include all off-diagonal terms between spin-orbitals and access antiferromagnetic ordering at strong coupling consitutes a powerful tool to determine the many-body nature of 1T-TaSe 2 .